*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        c(b(a(),a()),b(y,z),x) -> b(a(),b(z,z))
        f(b(b(x,f(y)),z)) -> c(z,x,f(b(b(f(a()),y),y)))
        f(c(c(a(),y,a()),b(x,z),a())) -> b(y,f(c(f(a()),z,z)))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {c/3,f/1} / {a/0,b/2}
      Obligation:
        Innermost
        basic terms: {c,f}/{a,b}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        c#(b(a(),a()),b(y,z),x) -> c_1()
        f#(b(b(x,f(y)),z)) -> c_2(c#(z,x,f(b(b(f(a()),y),y))),f#(b(b(f(a()),y),y)),f#(a()))
        f#(c(c(a(),y,a()),b(x,z),a())) -> c_3(f#(c(f(a()),z,z)),c#(f(a()),z,z),f#(a()))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        c#(b(a(),a()),b(y,z),x) -> c_1()
        f#(b(b(x,f(y)),z)) -> c_2(c#(z,x,f(b(b(f(a()),y),y))),f#(b(b(f(a()),y),y)),f#(a()))
        f#(c(c(a(),y,a()),b(x,z),a())) -> c_3(f#(c(f(a()),z,z)),c#(f(a()),z,z),f#(a()))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        c(b(a(),a()),b(y,z),x) -> b(a(),b(z,z))
        f(b(b(x,f(y)),z)) -> c(z,x,f(b(b(f(a()),y),y)))
        f(c(c(a(),y,a()),b(x,z),a())) -> b(y,f(c(f(a()),z,z)))
      Signature:
        {c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/0,c_2/3,c_3/3}
      Obligation:
        Innermost
        basic terms: {c#,f#}/{a,b}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        c#(b(a(),a()),b(y,z),x) -> c_1()
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        c#(b(a(),a()),b(y,z),x) -> c_1()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/0,c_2/3,c_3/3}
      Obligation:
        Innermost
        basic terms: {c#,f#}/{a,b}
    Applied Processor:
      Trivial
    Proof:
      Consider the dependency graph
        1:S:c#(b(a(),a()),b(y,z),x) -> c_1()
           
        
      The dependency graph contains no loops, we remove all dependency pairs.
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/0,c_2/3,c_3/3}
      Obligation:
        Innermost
        basic terms: {c#,f#}/{a,b}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).